# Sturm's Theorem

(redirected from Sturmian Sequence)

## Sturm's theorem

[′stərmz ‚thir·əm]
(mathematics)
This gives a method to determine the number of real roots of a polynomial p (x) which lie between two given values of x ; the Sturm sequence of p (x) provides the necessary information.

## Sturm’s Theorem

a theorem that provides a basis for finding nonoverlapping intervals such that each contains one real root of a given algebraic polynomial with real coefficients. The theorem was given in 1829 by J. C. F. Sturm.

For any polynomial f(x) without multiple roots, there exists a system of polynomials

f(x) = f0(x), f1(x),...,fs(x)

for which the following conditions are fulfilled: (1) fk(x) and fk+1(x), k = 0, 1,..., s – 1, do not have common roots; (2) the polynomial fs (x) has no real roots; (3) it follows from fk (α) = 0, 1 ≤ ks –1 that fk–1 (α)fk+1 (α) < 0; and (4) it follows from f(α) = 0 that the product f(x)f1(x) is increasing at the point α. Let w(c) be the number of changes of sign in the system

f(c), f1(c),...,fs(c)

If the real numbers a and b (a < b) are not roots of the polynomial f(x), then the difference w(a) – w(b) is nonnegative and equal to the number of real roots of the polynomial f(x) that lie between a and b. Thus, the number line may be divided into intervals each of which contains one real root of the polynomial f(x).

References in periodicals archive ?
upper) Sturmian sequence with slope a and intercept [rho].
And [tau] is called Sturmian if [tau]([xi]) is a Sturmian sequence for any Sturmian sequence [xi].
tau] ([xi]) = [xi]), then [xi] is a Sturmian sequence (with the following exceptions of non-primitive substitutions: ([01.
Which kind of Sturmian sequence can be fixed by certain non-trivial substitutions?
Yasutomi [10] gave a complete answer to the second question, by considering how the three elementary invertible substitutions change the slope and intercept of a Sturmian sequence.
Indeed, given a Sturmian sequence u [member of] [{0,1}.
Let us recall that [phi] is a Sturmian morphism, that is, for any Sturmian sequence u, the sequence y>(u) is Sturmian [Par97, MS93].
If 0w is Sturmian of type 0, then there exists a unique Sturmian sequence u satisfying [phi](u) = 0w.
There exist [alpha] [member of] {0,1}, [beta] [member of] N and a Sturmian sequence u of type 0 such that the sequence [0.
Christophe Reutenaur gave the first five lectures, on Christoffel words, which are finitary versions of Sturmian sequences.
In this section, we will present the notion of balanced sequences, which is closely related to the notion of Sturmian sequences [Lothaire 1991] as well as exactly covering sequences.
Intertwined periodic sequences, Sturmian sequences and Beatty sequence.
Site: Follow: Share:
Open / Close